A group of adults and kids went to see a movie. Tickets cost $$6.50$ each for adults and $$2.50$ each for kids, and the group paid $$30.50$ in total. There were $5$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${6.5x+2.5y = 30.5}$ ${x = y-5}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-5}$ for $x$ in the first equation. ${6.5}{(y-5)}{+ 2.5y = 30.5}$ Simplify and solve for $y$ $ 6.5y-32.5 + 2.5y = 30.5 $ $ 9y-32.5 = 30.5 $ $ 9y = 63 $ $ y = \dfrac{63}{9} $ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into ${x = y-5}$ to find $x$ ${x = }{(7)}{ - 5}$ ${x = 2}$ You can also plug ${y = 7}$ into ${6.5x+2.5y = 30.5}$ and get the same answer for $x$ ${6.5x + 2.5}{(7)}{= 30.5}$ ${x = 2}$ There were $2$ adults and $7$ kids.